All subjects · Further Mathematics · WFM02 Further Pure Mathematics 2

Using the standard Maclaurin series

Edexcel IAL Further Mathematics · WFM02 Further Pure Mathematics 2

Once you know the standard series, you can find many others without differentiating at all: substitute into them, multiply them, or combine them with the laws of logarithms.

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Substitution

Replace x by an expression: e-x2 = 1 - x2 + x4/2! - x6/3! + ...; cos 2x = 1 - (2x)2/2! + (2x)4/4! - ....

Put brackets round the substituted expression so powers are applied to everything: (2x)4 = 16x4.

Multiplying series

To expand ex sin x up to x3, multiply (1 + x + x2/2 + x3/6)(x - x3/6) and keep only terms up to x3: x + x2 + x3/2 - x3/6 = x + x2 + x3/3.

Logarithms

ln((1 + x)/(1 - x)) = ln(1 + x) - ln(1 - x) = 2(x + x3/3 + x5/5 + ...), valid for -1 < x < 1.

With x = 1/2 this gives a quickly converging series for ln 3.

Validity and uses

The range of validity changes with the substitution: ln(1 + 2x) needs -1 < 2x ≤ 1, so -½ < x ≤ ½.

Series give approximate values of integrals such as ∫ e-x2 dx and limits such as (1 - cos x)/x2 as x → 0.

Worked example

Find the series for ln((1 + x)/(1 - x)) up to x5, and use it to estimate ln 3.

  1. ln(1 + x) = x - x2/2 + x3/3 - x4/4 + x5/5 - ...
  2. ln(1 - x) = -x - x2/2 - x3/3 - x4/4 - x5/5 - ...
  3. Subtract: 2x + 2x3/3 + 2x5/5.
  4. (1 + x)/(1 - x) = 3 when x = 1/2: 2(0.5 + 0.0416... + 0.00625).

Answer: ln 3 ≈ 1.0958 (true value 1.0986)

Key idea: Substitute into, multiply or combine the standard series instead of differentiating. Check the new range of validity.

Check yourself

  1. Find the coefficient of x4 in the Maclaurin series for cos 2x. Give your answer as a fraction in its simplest form.
    Show answer

    2/3. (2x)4/4! = 16x4/24 = 2x4/3.

  2. Find the coefficient of x3 in the series for ex sin x. Give your answer as a fraction in its simplest form.
    Show answer

    1/3. From (1 + x + x2/2 + x3/6)(x - x3/6): the x3 terms are x3/2 - x3/6 = x3/3.

  3. Find the coefficient of x4 in the series for e-x2. Give your answer as a fraction in its simplest form.
    Show answer

    1/2. eu with u = -x2: the u2/2! term is x4/2.

  4. Use e-x2 ≈ 1 - x2 + x4/2 - x6/6 to estimate ∫ from 0 to 1 of e-x2 dx, to 4 decimal places.
    Show answer

    0.7429. Integrate term by term: 1 - 1/3 + 1/10 - 1/42 = 0.7429.

  5. For which values of x is the series for ln(1 + 2x) valid?
    • -1 < x ≤ 1
    • -2 < x ≤ 2
    • -½ < x ≤ ½
    • All real x
    Show answer

    -½ < x ≤ ½. ln(1 + u) needs -1 < u ≤ 1. With u = 2x: -½ < x ≤ ½.

  6. Use the series for cos x to find the limit of (1 - cos x)/x2 as x → 0. Give your answer as a fraction in its simplest form.
    Show answer

    1/2. 1 - cos x = x2/2 - x4/24 + ..., so (1 - cos x)/x2 = 1/2 - x2/24 + ... → 1/2.

  7. Find the coefficient of x4 in the series for cos(2x2).
    Show answer

    -2. cos u = 1 - u2/2 + ... with u = 2x2: u2/2 = 4x4/2 = 2x4, so the coefficient is -2.

  8. Using ln((1 + x)/(1 - x)) ≈ 2(x + x3/3 + x5/5) with x = 1/2, estimate ln 3 to 4 decimal places.
    Show answer

    1.0958. 2(0.5 + 0.125/3 + 0.03125/5) = 1.0958.

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Last updated 8 November 2026. Spotted a mistake? Tell us.